Question Video: Differentiating Combinations of Polynomial and Logarithmic Functions Using Product and Chain Rules | Nagwa Question Video: Differentiating Combinations of Polynomial and Logarithmic Functions Using Product and Chain Rules | Nagwa

# Question Video: Differentiating Combinations of Polynomial and Logarithmic Functions Using Product and Chain Rules Mathematics

Find the first derivative of the function π¦ = β7π₯β΄ ln 6π₯β΄.

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### Video Transcript

Find the first derivative of the function π¦ equals negative seven π₯ to the fourth power times the natural log of six π₯ to the fourth power.

Here, we have the product of two differentiable functions. The first is negative seven π₯ to the fourth power and the second is the natural log of six π₯ of the fourth power. We can, therefore, find the first derivative of our function by using the product rule. This says that the derivative of the product of two differentiable functions, π’ and π£, is π’ times dπ£ by dπ₯ plus π£ times dπ’ by dπ₯. If we let π’ be equal to negative seven π₯ to the fourth power, then dπ’ by dπ₯ is equal to four times negative seven π₯ cubed. Thatβs negative 28π₯ cubed. We then let π£ be equal to the natural log of six π₯ to the fourth power.

So how do we obtain dπ£ by dπ₯? Well, we can do one of two things. We could spot that we have a composite function and use the chain rule to find its derivative. Alternatively, we can quote the general result for the derivative of the logarithm of some function π of π₯. This says that if π¦ is equal to the natural log of this function π of π₯, then dπ¦ by dπ₯ is equal to π prime of π₯, the derivative of that function, divided by the original function π of π₯. In this case, π of π₯ is equal to six π₯ to the fourth power. So π prime of π₯, the derivative of this, is four times six π₯ cubed, which is 24π₯ cubed. dπ£ by dπ₯ is, therefore, 24π₯ cubed, divided by the original function, six π₯ to the fourth power.

Well, this simplifies quite nicely to four over π₯. And we can now substitute everything we have into the formula for the product rule. dπ¦ by dπ₯ is π’ times dπ£ by dπ₯. Thatβs negative seven π₯ to the fourth power times four over π₯ plus π£ times dπ’ by dπ₯. Thatβs the natural log of six π₯ to the fourth power times negative 28π₯ cubed. We simplify, and then we factor negative 28π₯ cubed. And we obtained dπ¦ by dπ₯ to be negative 28π₯ cubed times the natural logarithm six π₯ to the fourth power plus one.